Veronese parent of A109446 → (θ+1)²

nonnegative coefficients

Proved for all n all roots real, in (−∞, 0]coefficients of one sign Hadamard kernel

Recurrence

\(\left(-144 + 36\,x - 26\,n - 80\,n x + 170\,n^{2} - 64\,n^{2} x + 108\,n^{3} x\right)\,P_{n}(x) = \left(-144 - 540\,x + 144\,x^{2} + 82\,n - 18\,n x - 1116\,n x^{2} + 28\,n x^{3} + 72\,n x^{4} + 259\,n^{2} + 628\,n^{2} x + 578\,n^{2} x^{2} - 128\,n^{2} x^{3} + 56\,n^{2} x^{4} - 197\,n^{3} + 58\,n^{3} x + 493\,n^{3} x^{2} + 56\,n^{3} x^{3} - 93\,n^{3} x^{4} - 128\,n^{4} x - 55\,n^{4} x^{2} + 60\,n^{4} x^{3} - 42\,n^{4} x^{4} - 44\,n^{5} x^{2} - 8\,n^{5} x^{3} + 7\,n^{5} x^{4} - 8\,n^{6} x^{3}\right)\,P_{n-1}(x) + \left(-18 - 204\,x + 144\,x^{2} + 144\,x^{3} - 330\,x^{4} - 144\,x^{5} - 38\,n - 48\,n x - 508\,n x^{2} - 434\,n x^{3} - 48\,n x^{4} - 112\,n x^{5} + 56\,n^{2} + 200\,n^{2} x + 162\,n^{2} x^{2} + 92\,n^{2} x^{3} + 260\,n^{2} x^{4} + 186\,n^{2} x^{5} + 52\,n^{3} x + 146\,n^{3} x^{2} + 110\,n^{3} x^{3} + 100\,n^{3} x^{4} + 84\,n^{3} x^{5} + 56\,n^{4} x^{2} + 72\,n^{4} x^{3} + 2\,n^{4} x^{4} - 14\,n^{4} x^{5} + 16\,n^{5} x^{3} + 16\,n^{5} x^{4}\right)\,P_{n-1}'(x) + \left(4 - 40\,x - 140\,x^{2} - 302\,x^{3} - 436\,x^{4} - 302\,x^{5} - 72\,x^{6} - 3\,n - 24\,n x - 235\,n x^{2} - 632\,n x^{3} - 768\,n x^{4} - 478\,n x^{5} - 128\,n x^{6} + 28\,n^{2} x + 5\,n^{2} x^{2} - 166\,n^{2} x^{3} - 270\,n^{2} x^{4} - 162\,n^{2} x^{5} - 35\,n^{2} x^{6} - 12\,n^{3} x^{2} - 40\,n^{3} x^{3} - 37\,n^{3} x^{4} - 2\,n^{3} x^{5} + 7\,n^{3} x^{6} - 8\,n^{4} x^{3} - 16\,n^{4} x^{4} - 8\,n^{4} x^{5}\right)\,P_{n-1}''(x)\)

\(P_{0}(x) = 4\,x\)

\(P_{1}(x) = 1 + 4\,x\)

Rows ?

n	x⁰	x¹	x²	x³	x⁴	x⁵	x⁶	x⁷	x⁸	x⁹	x¹⁰	x¹¹	x¹²	x¹³	x¹⁴	x¹⁵	P_n(1)
0	0	4															4
1	1	4															5
2	0	4	18	16													38
3	1	12	27	16													56
4	0	4	36	96	100	36											272
5	1	20	90	160	125	36											432
6	0	4	54	240	500	540	294	64									1696
7	1	28	189	560	875	756	343	64									2816
8	0	4	72	448	1400	2520	2744	1792	648	100							9728
9	1	36	324	1344	3150	4536	4116	2304	729	100							16640
10	0	4	90	720	3000	7560	12348	13440	9720	4500	1210	144					52736
11	1	44	495	2640	8250	16632	22638	21120	13365	5500	1331	144					92160
12	0	4	108	1056	5500	17820	38808	59136	64152	49500	26620	9504	2028	196			274432
13	1	52	702	4576	17875	46332	84084	109824	104247	71500	34606	11232	2197	196			487424
14	0	4	126	1456	9100	36036	98098	192192	277992	300300	242242	144144	61516	17836	3150	256	1384448

Technical details

Node id: n-4971b95a96ff967e · revision 0 · JSON
Definition source: derived:theta_plus_one_squared
Fingerprint: f6-18673a984a1fab0a

Observed on the stored rows ?

Degrees: 1, 1, 3, 3, 5, 5, 7, 7, 9, 9, 11, 11, 13, 13, 15 (irregular)
Palindromic rows: no
γ-positive (gamma_positive): no
Nonnegative coefficients (nonn): yes
Only rational roots (rr_by_factors: every nonzero row is a product of linear factors over ℚ): no

Matches ?

No OEIS matches.

Submission text ?

Atlas rows n = 0, …, ascending powers of x

%e Example

Triangle begins:
  0: 0,  4;
  1: 1,  4;
  2: 0,  4,  18,   16;
  3: 1, 12,  27,   16;
  4: 0,  4,  36,   96,  100,   36;
  5: 1, 20,  90,  160,  125,   36;
  6: 0,  4,  54,  240,  500,  540,  294,   64;
  7: 1, 28, 189,  560,  875,  756,  343,   64;
  8: 0,  4,  72,  448, 1400, 2520, 2744, 1792, 648, 100;
  9: 1, 36, 324, 1344, 3150, 4536, 4116, 2304, 729, 100;

%F Formula

The row polynomials satisfy (170*n^2-26*n-144+(108*n^3-64*n^2-80*n+36)*x)*P(n,x) = -(197*n^3-259*n^2-82*n+144+(128*n^4-58*n^3-628*n^2+18*n+540)*x+(44*n^5+55*n^4-493*n^3-578*n^2+1116*n-144)*x^2+(8*n^6+8*n^5-60*n^4-56*n^3+128*n^2-28*n)*x^3-(7*n^5-42*n^4-93*n^3+56*n^2+72*n)*x^4)*P(n-1,x) + (56*n^2-38*n-18+(52*n^3+200*n^2-48*n-204)*x+(56*n^4+146*n^3+162*n^2-508*n+144)*x^2+(16*n^5+72*n^4+110*n^3+92*n^2-434*n+144)*x^3+(16*n^5+2*n^4+100*n^3+260*n^2-48*n-330)*x^4-(14*n^4-84*n^3-186*n^2+112*n+144)*x^5)*d/dx P(n-1,x) - (3*n-4-(28*n^2-24*n-40)*x+(12*n^3-5*n^2+235*n+140)*x^2+(8*n^4+40*n^3+166*n^2+632*n+302)*x^3+(16*n^4+37*n^3+270*n^2+768*n+436)*x^4+(8*n^4+2*n^3+162*n^2+478*n+302)*x^5-(7*n^3-35*n^2-128*n-72)*x^6)*d^2/dx^2 P(n-1,x) for n >= 2, with P(0,x) = 4*x, P(1,x) = 1 + 4*x.

%t Mathematica

P[0, x_] := 4*x;
P[1, x_] := 1 + 4*x;
P[n_, x_] := P[n, x] = Expand[Together[(-(197*n^3 - 259*n^2 - 82*n + 144 + (128*n^4 - 58*n^3 - 628*n^2 + 18*n + 540)*x + (44*n^5 + 55*n^4 - 493*n^3 - 578*n^2 + 1116*n - 144)*x^2 + (8*n^6 + 8*n^5 - 60*n^4 - 56*n^3 + 128*n^2 - 28*n)*x^3 - (7*n^5 - 42*n^4 - 93*n^3 + 56*n^2 + 72*n)*x^4)*P[n - 1, x] + (56*n^2 - 38*n - 18 + (52*n^3 + 200*n^2 - 48*n - 204)*x + (56*n^4 + 146*n^3 + 162*n^2 - 508*n + 144)*x^2 + (16*n^5 + 72*n^4 + 110*n^3 + 92*n^2 - 434*n + 144)*x^3 + (16*n^5 + 2*n^4 + 100*n^3 + 260*n^2 - 48*n - 330)*x^4 - (14*n^4 - 84*n^3 - 186*n^2 + 112*n + 144)*x^5)*D[P[n - 1, x], x] - (3*n - 4 - (28*n^2 - 24*n - 40)*x + (12*n^3 - 5*n^2 + 235*n + 140)*x^2 + (8*n^4 + 40*n^3 + 166*n^2 + 632*n + 302)*x^3 + (16*n^4 + 37*n^3 + 270*n^2 + 768*n + 436)*x^4 + (8*n^4 + 2*n^3 + 162*n^2 + 478*n + 302)*x^5 - (7*n^3 - 35*n^2 - 128*n - 72)*x^6)*D[P[n - 1, x], {x, 2}])/(170*n^2 - 26*n - 144 + (108*n^3 - 64*n^2 - 80*n + 36)*x)]];
Table[CoefficientList[P[n, x], x], {n, 0, 9}] // Flatten

%o PARI/GP

P(N) = {my(v = vector(N + 1)); v[1] = 4*x; v[2] = 1 + 4*x; for(n = 2, N, my(i = n + 1); v[i] = (-(197*n^3 - 259*n^2 - 82*n + 144 + (128*n^4 - 58*n^3 - 628*n^2 + 18*n + 540)*x + (44*n^5 + 55*n^4 - 493*n^3 - 578*n^2 + 1116*n - 144)*x^2 + (8*n^6 + 8*n^5 - 60*n^4 - 56*n^3 + 128*n^2 - 28*n)*x^3 - (7*n^5 - 42*n^4 - 93*n^3 + 56*n^2 + 72*n)*x^4)*v[i - 1] + (56*n^2 - 38*n - 18 + (52*n^3 + 200*n^2 - 48*n - 204)*x + (56*n^4 + 146*n^3 + 162*n^2 - 508*n + 144)*x^2 + (16*n^5 + 72*n^4 + 110*n^3 + 92*n^2 - 434*n + 144)*x^3 + (16*n^5 + 2*n^4 + 100*n^3 + 260*n^2 - 48*n - 330)*x^4 - (14*n^4 - 84*n^3 - 186*n^2 + 112*n + 144)*x^5)*deriv(v[i - 1], x) - (3*n - 4 - (28*n^2 - 24*n - 40)*x + (12*n^3 - 5*n^2 + 235*n + 140)*x^2 + (8*n^4 + 40*n^3 + 166*n^2 + 632*n + 302)*x^3 + (16*n^4 + 37*n^3 + 270*n^2 + 768*n + 436)*x^4 + (8*n^4 + 2*n^3 + 162*n^2 + 478*n + 302)*x^5 - (7*n^3 - 35*n^2 - 128*n - 72)*x^6)*deriv(deriv(v[i - 1], x), x))/(170*n^2 - 26*n - 144 + (108*n^3 - 64*n^2 - 80*n + 36)*x)); v};
row(n, p) = {my(c = Vecrev(p)); c};
T(N) = {my(v = P(N)); concat(vector(#v, i, row(i - 1, v[i])))};
T(9)

polytool c48629374597

Properties

Real-rooted? Every row P_n, n ≥ 0, is zero or real-rooted with all roots in (−∞, 0]
by Laguerre’s theorem on multiplier sequences
Coefficients of one sign? Every nonzero row, n ≥ 0: the coefficients of P_n have one sign
derived from Laguerre’s theorem on multiplier sequences
Hadamard kernel (Malo)? Rows P_n, n ≥ 0, are Hadamard kernels (preserve real-rootedness)
derived from Laguerre’s theorem on multiplier sequences

Proof chains ? JSON

Laguerre’s theorem on multiplier sequences [SP14; Bra15, Thm 7.1 of arXiv:1410.6601] polytool ? step ›
- from veronese parent of Binomial coefficients C(n,k) with n-k even, read by rows.?
  - Iterated real-rootedness preservers [SP14; Bra15, Thm 7.1 of arXiv:1410.6601; Bra06, Cor 3.5] polytool ? step › direct check
  - Parent-row provers of polytool polytool ? step › direct check

References

[Bra06] Petter Brändén. On linear transformations preserving the Pólya frequency property. Transactions of the American Mathematical Society, 358(8):3697–3716, February 2006. doi:10.1090/s0002-9947-06-03856-6
[Bra15] Petter Brändén. Unimodality, log-concavity, real-rootedness and beyond. Handbook of enumerative combinatorics, :437–483, March 2015. doi:10.1201/b18255-10
[SP14] J Schur and G Polya. Über zwei Arten von Faktorenfolgen in der Theorie der algebraischen Gleichungen. J. Reine Angew. Math., 1914.